Horsetail is practically impossible to get rid of as the roots go down many meters. Chemicals in a number of trials have had limited success however use in a vegetable garden would not be advisable. One method that I know has worked is to grass down your vegetable patch cutting regularly with a lawn mower for between one and two years. Regular mowing has the effect of weakening the horsetail and eventually killing it.
You may just have enough time before the cold weather sets in to lay some turf or wait untill the spring, however in the spring the horsetail will be springing into growth.
I have never heard of growing parsnips to eradicate horsetail and think it unlikely to work due to the persistent nature of horsetail. Good luck